What is an eigenmatrix and what is its use? I stumbled upon a paper using the term eigenmatrix which I never heard of before. Sadly I found little to none literature to it. Even my books at home don't know the term.
I believe that an eigenmatrix $E$ of some $(n\times n)$ matrix $A$ with eigenvalues $\lambda_1,\ldots,\lambda_n$ is of form:
$E=\begin{align}\left( \begin{array}{rrrr}
\lambda_1 & 0 & 0 \\
0 & \ddots & 0  \\
0 & 0 & \lambda_n 
\end{array}\right)\end{align} $ where each $\lambda_i$ is represented as often as its multiplicity.
If my definition up to this point is correct, my questions are:

*

*What is this matrix used for?

*How does it look like if $A$ doesn't have a full set of (real)
eigenvalues?

*Do eigenvectors play any role, if so what? (I ask this primarily because the paper I read states that "$E$ is an eigenmatrix of $A$ corresponding to an eigenvector.")

*How can one calculate an eigenmatrix of some square matrix $A$?

If my definition wasn't correct, feel free to tell me what an eigenmatrix is.
 A: The only eigenmatrix application that I know is it's use in Independent Component Analysis (ICA). There is an algorithm for ICA which is called Joint Approximate Diagonalization of Eigenmatrices (JADE) which diagonalizes a 4-dimentional cumulant $N\times N\times N\times N$ tensor $\boldsymbol Q$ by the diagonalization of it's eigenmatrices where an eigenmatrix is a $N\times N$ matrix defined as:
$$\boldsymbol F(\boldsymbol M)=\lambda\boldsymbol M$$
where $\boldsymbol F(\boldsymbol M)$ is a linear transformation of a matrix defined by the 4-dimentional tensor as:
$$[\boldsymbol F(\boldsymbol M)]_{ij}=\sum_{ij}m_{kl}q_{ijkl}$$
In this case, the eigenmatrix of a 4D tensor can be easely found by any standard eigendecomposition algorithm if you reshape the $N\times N\times N\times N$ tensor to a $N^2\times N^2$ matrix then find it's $N^2$-dimentional eigenvectors which can be reshaped to $N\times N$ matrices and became the eigenmatrices of the 4D tensor.
More details can be found on chapter 11 of this book
